Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
What is h-homogeneity?
Overview of the results
The tools
Products and h-homogeneity
Andrea Medini
Department of Mathematics
University of Wisconsin - Madison
June 27, 2010
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
What is h-homogeneity?
Overview of the results
The tools
All spaces we consider are Tychonoff (so that we can take the
Stone-Čech compactification) and infinite.
A space X is zero-dimensional if it has a T1 basis consisting of
clopen sets. Every such space is Tychonoff.
Definition (Ostrovskii, 1981; van Mill, 1981)
A topological space X is h-homogeneous (or strongly
homogeneous) if all non-empty clopen sets in X are
homeomorphic.
Examples:
The Cantor set 2ω , the rationals Q, the irrationals ω ω . (Use
the respective characterizations.)
Any connected space.
The Knaster-Kuratowski fan.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
What is h-homogeneity?
Overview of the results
The tools
Main results
In the class of zero-dimensional spaces, h-homogeneity is
productive.
If the product is pseudocompact, then the
zero-dimensionality requirement can be dropped.
Clopen sets in pseudocompact products depend on finitely
many coordinates.
Partial answers to Terada’s question: is the infinite power
X ω h-homogeneous for every zero-dimensional
first-countable X ?
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
What is h-homogeneity?
Overview of the results
The tools
A useful base for βX
Definition
Given U open in X , define Ex(U) = βX \ clβX (X \ U).
Basic facts:
Ex(U) is the biggest open set in βX such that its
intersection with X is U.
The collection {Ex(U) : U is open in X } is a base for βX .
If C is clopen in X then Ex(C) = clβX (C), hence Ex(C) is
clopen in βX .
j It is not true that βX
is zero-dimensional whenever X is
zero-dimensional. (Dowker, 1957.) j
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
What is h-homogeneity?
Overview of the results
The tools
Q
When does β commute with
?
Theorem (Glicksberg, 1959)
Q
Q
The product i∈I Xi is C ∗ -embedded in i∈I βXi if and only if
Q
i∈I Xi is pseudocompact.
In that case,
!
Y
βXi ∼
=β
i∈I
Y
Xi
.
i∈I
More precisely, there exists a homeomorphism
!
Y
Y
h:
βXi −→ β
Xi
i∈I
such that h Q
i∈I
i∈I
Xi = id.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
The productivity of h-homogeneity
Theorem (Terada, 1993)
If Xi is h-homogeneous
and zero-dimensional for every i ∈ I
Q
and P = i∈I Xi is compact or non-pseudocompact, then P is
h-homogeneous.
Proof of the compact case, for P = X × Y :
Observe that n × X ∼
= X whenever 1 ≤ n < ω.
∼
So n × X × Y = X × Y whenever 1 ≤ n < ω.
Let C be non-empty and clopen in X × Y . By compactness,
zero-dimensionality and Q, find clopen rectangles Ci such that
C = C1 ⊕ · · · ⊕ Cn .
By h-homogeneity, C ∼
=n×X ×Y ∼
= X × Y.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Theorem
If X × Y is pseudocompact, then every clopen set C can be
written as a finite union of open rectangles.
Proof: By Glicksberg’s theorem, there exists a homeomorphism
h : βX × βY −→ β(X × Y )
such that h(x, y ) = (x, y ) whenever (x, y ) ∈ X × Y .
Since {Ex(U) : U is open in X } is a base for βX and
{Ex(V ) : V is open in Y } is a base for βY , the collection
B = {Ex(U) × Ex(V ) : U is open in X and V is open in Y }
is a base for βX × βY .
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Therefore {h[B] : B ∈ B} is a base for β(X × Y ).
Hence we can write Ex(C) = h[B1 ] ∪ · · · ∪ h[Bn ] for some
B1 , . . . , Bn ∈ B by compactness.
Finally, if we let Bi = Ex(Ui ) × Ex(Vi ) for each i, we get
C = Ex(C) ∩ X × Y
= (h[B1 ] ∪ · · · ∪ h[Bn ]) ∩ h[X × Y ]
= h[B1 ∩ X × Y ] ∪ · · · ∪ h[Bn ∩ X × Y ]
= (B1 ∩ X × Y ) ∪ · · · ∪ (Bn ∩ X × Y )
= (U1 × V1 ) ∪ · · · ∪ (Un × Vn ).
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
But we would like clopen rectangles... /
Why? Because then we could prove the following.
(Notice that zero-dimensionality is not needed.)
Theorem
Assume that X × Y is pseudocompact. If X and Y are
h-homogeneous then X × Y is h-homogeneous.
Proof: If X and Y are both connected then X × Y is connected,
so assume without loss of generality that X is not connected.
It follows that X ∼
= n × X whenever 1 ≤ n < ω.
...then finish the proof as in the compact case.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Lemma
Let C ⊆ X × Y be a clopen set that can be written as the union
of finitely many rectangles. Then C can be written as the union
of finitely many pairwise disjoint clopen rectangles. ,
[ . Draws an enlightening picture on the board.]
Proof: For every x ∈ X , let Cx = {y ∈ Y : (x, y ) ∈ C} be the
corresponding vertical cross-section. For every y ∈ Y , let
C y = {x ∈ X : (x, y ) ∈ C} be the corresponding horizontal
cross-section. Since C is clopen, each cross-section is clopen.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Let A be the Boolean subalgebra of the clopen algebra of X
generated by {C y : y ∈ Y }. Since A is finite, it must be atomic.
Let P1 , . . . , Pm be the atoms of A. Similarly, let B be the
Boolean subalgebra of the clopen algebra of Y generated by
{Cx : x ∈ X }, and let Q1 , . . . , Qn be the atoms of B.
Observe that the rectangles Pi × Qj are clopen and pairwise
disjoint. Furthermore, given any i, j, either Pi × Qj ⊆ C or
Pi × Qj ∩ C = ∅. Hence C is the union of a (finite) collection of
such rectangles.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Corollary
Assume that X = X1 × · · · × Xn is pseudocompact. If each Xi is
h-homogeneous then X is h-homogeneous.
An obvious modification of the proof of the theorem yields:
Theorem
Q
Assume that X = i∈I Xi is pseudocompact. If C ⊆ X is clopen
then C can be written as the union of finitely many open
rectangles.
Corollary
Q
Assume that X = i∈I Xi is pseudocompact. If C ⊆ X is clopen
then C depends on finitely many coordinates.
[The speaker takes a walk down memory lane...]
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Theorem
Q
Assume that X = i∈I Xi is pseudocompact. If Xi is
h-homogeneous for every i ∈ I then X is h-homogeneous.
Proof: Let C ⊆ X be clopen and non-empty.
Then there exists a finite
that C is
Qsubset F of I such
0×
0 is a clopen subset
homeomorphic
to
C
X
,
where
C
i
i∈I\F
Q
of Q
i∈F Xi .
But i∈F Xi is h-homogeneous, so
C∼
= C0 ×
Y
Xi ∼
=
i∈I\F
Y
i∈F
Xi0 ×
Y
Xi ∼
= X.
i∈I\F
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Conclusions
Putting together our results with Terada’s theorem, we obtain
the following.
Theorem
If Xi is h-homogeneous
and zero-dimensional for every i ∈ I
Q
and X = i∈I Xi then X is h-homogeneous.
After all this work...
Problem
Is h-homogeneity productive?
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
Some applications
The following result was first proved by Motorov in the compact
case.
Theorem
Assume that
Q X has a π-base B consisting of clopen sets. Then
(X × 2 × B)κ is h-homogeneous for every infinite cardinal κ.
Corollary
For every zero-dimensional space X there exists a
zero-dimensional space Y such that X × Y is h-homogeneous.
Problem
Is it true that for every space X there exists a space Y such that
X × Y is h-homogeneous?
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
The proof
Some applications
The case κ = ω of the following result is an easy consequence
of a result of Matveev. Motorov first proved it under the
additional assumption that X is first-countable and compact.
Terada proved it for an arbitrary infinite κ, under the additional
assumption that X is non-pseudocompact.
Theorem
Assume that X is a space such that the isolated points are
dense in X . Then X κ is h-homogeneous for every infinite
cardinal κ.
For example, if α is an ordinal with the order topology and κ is
an infinite cardinal then ακ is h-homogeneous.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Homogeneity vs h-homogeneity
All spaces are assumed to be first-countable and
zero-dimensional from now on.
Definition
A space X is homogeneous if for every x, y ∈ X there exists a
homeomorphism f : X −→ X such that f (x) = y .
By a picture-proof, h-homogeneity implies homogeneity. .
Erik van Douwen constructed a compact homogeneous space
that is not h-homogeneous.
Theorem (Motorov, 1989)
If X is a compact homogeneous space of uncountable
cellularity then X is h-homogeneous.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Infinite powers
Problem (Terada, 1993)
Is X ω always h-homogeneous?
The following remarkable theorem is based on work by Motorov
and Lawrence.
Theorem (Dow and Pearl, 1997)
X ω is homogeneous.
However, Terada’s question remains open.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Motorov’s main result
Theorem (Motorov, 1989)
If X has a π-base consisting of clopen sets that are
homeomorphic to X then X is h-homogeneous.
Proof: Let C be a non-empty clopen set in X . By
first-countability, write
[
[
X = {x} ∪
Xn
and
C = {y } ∪
Cn
n∈ω
n∈ω
where the Xn are disjoint, clopen, they converge to x but do not
contain x, and the Cn are disjoint, clopen, they converge to y
but do not contain y .
[ . Finishes the proof by juggling with clopen sets.]
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Divisibility
Definition
A space F is a factor of X (or X is divisible by F ) if there exists
Y such that F × Y ∼
= X . If F × X ∼
= X then F is a strong factor
of X (or X is strongly divisible by F ).
Problem (Motorov, 1989)
Is X ω always divisible by 2?
As we observed already, h-homogeneity implies divisibility by 2.
We will show that Terada’s question is equivalent to Motorov’s
question. Actually, even weaker conditions suffice.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Lemma
The following are equivalent.
1
2
3
F is a factor of X ω .
∼ X ω.
F × Xω =
Fω × Xω ∼
= X ω.
The implications 2 → 1 and 3 → 1 are clear.
Assume 1. Then there exists Y such that F × Y ∼
= X ω , hence
Xω ∼
= (F × Y )ω ∼
= F ω × Y ω.
= (X ω )ω ∼
Since multiplication by F or by F ω does not change the right
hand side, it follows that 2 and 3 hold.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
The key lemma
Lemma
X = (Y ⊕ 1)ω is h-homogeneous.
Proof: Recall that 1 = {0}. For each n ∈ ω, define
Un = {0} × {0} × · · · × {0} ×(Y ⊕ 1) × (Y ⊕ 1) × · · ·
|
{z
}
n times
Observe that {Un : n ∈ ω} is a local base for X at (0, 0, . . .)
consisting of clopen sets that are homeomorphic to X .
But X is homogeneous by the Dow-Pearl theorem, therefore it
has a base (hence a π-base) consisting of clopen sets that are
homeomorphic to X .
It follows from Motorov’s result that X is h-homogeneous.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Lemma
Let X = (Y ⊕ 1)ω . Then
X ∼
= Y ω × (Y ⊕ 1)ω ∼
= 2ω × Y ω .
Proof: Observe that
X ∼
= (Y ⊕ 1) × X ∼
= (Y × X ) ⊕ X ,
hence X ∼
= Y × X by h-homogeneity. It follows that
X ∼
= Y ω × (Y ⊕ 1)ω . Finally,
ω
Y ω × (Y ⊕ 1)ω ∼
= (Y ω ⊕ Y ω )ω ∼
= 2ω × Y ω ,
= (Y ω × (Y ⊕ 1)) ∼
that concludes the proof.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Theorem
The following are equivalent.
1
Xω ∼
= (X ⊕ 1)ω .
2
Xω ∼
= Y ω for some Y with at least one isolated point.
3
X ω is h-homogeneous.
4
X ω has a clopen subset that is strongly divisible by 2.
5
X ω has a proper clopen subspace homeomorphic to X ω .
6
X ω has a proper clopen subspace as a factor.
Proof: The implication 1 → 2 is trivial; the implication 2 → 3
follows from the lemma; the implications 3 → 4 → 5 → 6 are
trivial.
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Assume that 6 holds. Let C be a proper clopen subset of X ω
that is also a factor of X ω and let D = X ω \ C. Then
Xω ∼
= (C ⊕ D) × X ω
∼
= (C × X ω ) ⊕ (D × X ω )
∼
= X ω ⊕ (D × X ω )
∼
= (1 ⊕ D) × X ω ,
hence X ω ∼
= (1 ⊕ D)ω × X ω . Since (1 ⊕ D)ω ∼
= 2ω × D ω by the
lemma, it follows that X ω ∼
= 2ω × X ω . Therefore 1 holds by the
lemma.
K
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
The pseudocompact case
The next two theorems show that in the pseudocompact case
we can say something more.
Theorem
Assume that X ω is pseudocompact. Then C ω ∼
= (X ⊕ 1)ω for
ω
every non-empty proper clopen subset C of X .
Theorem
Assume that X ω is pseudocompact. Then the following are
equivalent.
1
2
X ω is h-homogeneous.
X ω has a proper clopen subspace C such that C ∼
= Y ω for
some Y .
Andrea Medini
Products and h-homogeneity
Preliminaries
The productivity of h-homogeneity
Infinite powers of zero-dimensional first-countable spaces
Homogeneity vs h-homogeneity
The equivalence of Terada’s and Motorov’s questions
Bonus materials
Ultraparacompactness
The following notion allows us to give us a positive answer to
Terada’s question for a certain class of spaces.
Definition
A space X is ultraparacompact if every open cover of X has a
refinement consisting of pairwise disjoint clopen sets.
A metric space X is ultraparacompact if and only if dim X = 0.
Theorem
If X ω is ultraparacompact and non-Lindelöf then X ω is
h-homogeneous.
Andrea Medini
Products and h-homogeneity